Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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1
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2answers
55 views

Use Dominated convergence theorem to show that $f(x):=\sum_{k=1}^{\infty}\frac{\cos(kx)}{k^3}$ is differentiable

Let $$f(x):=\sum_{k=1}^{\infty}\frac{\cos(kx)}{k^3},$$ how can we show that f is differentiable everywhere by using the Lebesgue dominated convergence theorem? I know this theorem as saying ...
2
votes
1answer
17 views

Reversing minimax function

Let $$g(x) = \inf_{a \in A} \sup_{b \in B} f(a,b,x).$$ When it is true that $$g^{-1}(x) = \inf_{a \in A} \sup_{b \in B} f^{-1}(a,b,x)\ ?$$ where $f^{-1}(a,b,x)$ means that $f^{-1}(a,b,f(a,b,x)) =x$ ...
0
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2answers
42 views

Prove a set of continuous function is bounded.

Let $C_b(\mathbb{R})$ denote the space of all bounded, continuous functions $f : \mathbb{R} → \Bbb C$. Let $C_0(\mathbb{R})$ denote the set of continuous functions $f : \mathbb{R} \to \Bbb C$ for ...
2
votes
2answers
54 views

How do I show that as $z \to \infty$ that $\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt = O(z^{-1} )$??

How do I show that as $z \to \infty$ that $\int_0^\infty \frac{t - \lfloor t \rfloor - 1/2}{z + t} dt = O(z^{-1} )$? According to Serge Lang, the integral on the left is the error term for Stirling's ...
5
votes
2answers
80 views

Finding solutions to $ x^x = 2x$

A friend claims it isn't possible to find a closed form for the smaller positive real solution of $x^x = 2x$. Numerically we have seen that $0.346...$ and $2$ are solutions, but are failing to do ...
1
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2answers
47 views

If $A$ has a maximum, prove that it only has one.

Let $A\subseteq \mathbb{R}$. We say that a real number $M\in\mathbb{R}$ is a maximum of $A$ if $M$ is an upper bound for $A$ and $M\in A$. If $A$ has a maximum, prove that it only has one; and prove ...
0
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0answers
18 views

Radial function and integral

Let $\Phi:(0,\infty)\rightarrow(0,\infty)$ be an increasing function and $\rho:(0,\infty)\rightarrow(0,\infty)$ is a function satisying the property $$ \frac{1}{C}\leq\frac{\rho(s)}{\rho(r)}\leq C ...
2
votes
1answer
52 views

Is this function Riemann integrable in $[0,1]$?

The function is $f(x) = 1$ for $ 0 \le x \lt 1 $ and $f(x) = 2$ for $x = 1$ I calculate the upper sum $$U(P,f) = \sum_{i=1}^n M_i \Delta x_i = \sum_{i=1}^{n-1} 1\,\Delta x_i + 2 \,\Delta x_n = ...
1
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0answers
37 views

How do you multiply or add two numbers with infinite number of digits? [closed]

I was reading an analysis book and I was confused about the section where the author explained how to multiply or add two numbers with infinite number of digits. Are there any intuitive thoughts on ...
0
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1answer
17 views

Is there a Heine cretierion of liminf of a function?

Lately i've been struggling with understanding the meaning of $\liminf_{x\to x_0}f(x)$ assuming $f:X\to\mathbb C$ for $X$ a metric space, or for that matter $f:\mathbb R\to \mathbb R$. Could you give ...
2
votes
1answer
77 views

Prove that there are infinitely many real numbers.

Here it goes: Assume that there is an upper bound first for the set $\mathbb{R}$ let $\alpha = \sup \mathbb{R}$ So assume $\alpha = \sup \mathbb{R}$ Therefore $\alpha \ge x$ for $x \in \mathbb{R}$ ...
2
votes
0answers
43 views

Boundedness of linear operators in $L^p$

I was wondering if the following conditions are equivalent for a linear operator $T$ between $L^p$ spaces: i) T maps $L^p$ to $L^p$, that is, if $f \in L^p$ then $Tf \in L^p$. ii) There exists ...
3
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0answers
122 views

Prove $\int_{0}^{1} |\frac {f^{''}(x)}{f(x)}| dx \geq 4$ [on hold]

I find an interesting theorem,but have no idea to prove it. $f(x) \in C^2[0,1]$ and $f(0)=f(1)=0$ , $f(x) \not = 0 \ \ , x\in (0,1) $ Prove that if $\int_{0}^{1} \bigl|\frac ...
1
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1answer
30 views

borel-measurable function is pointwise limit of a sequence of continuous functions, wich is uniformly bounded

Let $H$ be a Hilbert space over $\mathbb{C}$, $A\in L(H)$ ( $A:H\to H$ is linear and continuous) and let $A$ be self-adjoint. Consider the spectrum of A, $\sigma(A)$ and $f:K\to \mathbb{K}$ a bounded, ...
1
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1answer
37 views

Let $x=0,a_1a_2a_3\cdots a_i\cdots$ be a number such that $a_1=0$, $a_i=1$ if $i\in\mathbb{N}$ is a prime number and $a_i=0$ otherwise.

Let $x=0,a_1a_2a_3\cdots a_i\cdots$ be a number such that $a_1=0$, $a_i=1$ if $i\in\mathbb{N}$ is a prime number and $a_i=0$ otherwise. So $x=0,01101010001\cdots$. Is $x$ a rational number? How can I ...
2
votes
1answer
38 views

$f \in \mathcal{L}^1(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R}),f' \in \mathcal{L}^1(\mathbb{R}) \Longrightarrow f \in \mathcal{C}_{0}(\mathbb{R})$

I want to show the following theorem: Suppose $f \in \mathcal{L}^1(\mathbb{R}) \cap \mathcal{C}^1(\mathbb{R})$ and $f' \in \mathcal{L}^1(\mathbb{R})$. Then it holds that $f \in ...
0
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0answers
34 views

Are Piecewise continuous functions Lipschitz?

Are Piecewise continuous functions Lipschitz continuous as well? I understand that Lipschitz continuous $\implies$ absolutely continuous, and I think Piecewise continuous functions can be thought of ...
0
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0answers
40 views

Prove that $F = \{ x+\sqrt y \mid x, y \in \mathbb Q \}$ is an ordered field without the completeness property

Let $F$ be the set of all numbers of the form $x+\sqrt{y}$, where $x$ and $y$ are rational. Show that $F$ has all the properties of an ordered field but does not have the completeness property? How ...
1
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1answer
17 views

Showing $\sum_{j \geq 2}\sum_{x^{1/j} \leq p \leq x}\frac{1}{jp^{j}} = O\left(\frac{1}{\log x}\right)$

Let $p$ denote a prime. Suppose I am given the asymptotic that $$\sum_{1 \leq n \leq x} \frac{\Lambda(n)}{n\log n} = \log\log x + \gamma + O\left(\frac{1}{\log x}\right),$$ why is $$\sum_{2 \leq p ...
0
votes
2answers
41 views

if $S=\left\{ x \in Q | x^2 < 2\right\}$ Prove $S$ has an upper bound

So I claim that it is true because our professor believes so, and here is my proof, Suppose there is a number $x'> 2$ such that $x'^2 > 4$ however $x'^2 < 2$ therefore contradiction and ...
0
votes
1answer
28 views

Differential operator a bounded operator or not?

Is the operator $T$ a bounded operator mapping $T: H^n([0,\pi]) \rightarrow H^{n-1}([0,\pi])$ ($H^n$ is the n-th Sobolev space with respect to $L^2$) or not? The operator itself is given by ...
0
votes
1answer
10 views

how should a unbounded integrable function be like on a bounded set?

actually, my first question is could it be unbounded near boundary and we redefine the value on the boundary. For example, function $f=1/\sqrt{x}$ on $(0,1]$ and $f=0$ if $x=0$. is it integrable? Is ...
0
votes
1answer
34 views

Proof, that helical surface is a submainfold [closed]

I have to proof, that helical surface $M:= \left|\begin{array}{ccc}s\cos(t)\\s \sin (t)\\t\end{array}\right|$ s,t$\in R$ is 2 dimensional submanifold. How to do it?
0
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0answers
16 views

definition of open set and is gauge function is well defined?

Q1 :I know that a set A $ \subset$ X is open if it contains an open ball about each of its points i.e. for all x in A ,there exists $ \epsilon $>0 s.t. $ B_\epsilon $(x) $\subset $ A But then does ...
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0answers
23 views

Do you know any example borel algebra? [closed]

Example of a family $(Ωα)α∈[0,2π]$ of $σ$-subalgebras of the Borel algebra $B([0, 1]2)$ such that $\forall α, β ∈ [0, 2π], α 6= β: Ωα ∩ Ωβ = \{∅, [0, 1]2 \}$ and $(∀α)[α ∈ [0, π]](∀a)[0 ≤ a ≤ 1](∃B ∈ ...
5
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0answers
63 views

Examples of categorical adjunctions in analysis and differential geometry?

In a lot of introductory texts on category theory, it seems like the majority of examples come from algebraic topology, algebra, and logic. Are there any good examples of adjunctions in analysis and ...
5
votes
1answer
95 views
+100

Intuition behind functional dependence

What is the intuition behind functional independence ? (This is defined in the following way: Let $k\leq n$. The $C^1$ functions $F_1,\ldots,F_k:\mathbb{R}^n\rightarrow \mathbb{R}$ are functionally ...
0
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0answers
22 views

Relations between $\varepsilon$ and $\delta$ in the $\varepsilon-\delta$-Defintion of Continuity

A function $f : \mathbb R \to \mathbb R$ is continuous at $x_0$ iff for each $\varepsilon > 0$ there exists some $\delta > 0$ such that $$ |x - x_0| < \delta \Rightarrow |f(x) - f(x_0)| < ...
7
votes
5answers
684 views

Issue with Spivak's Solution

Here was the problem: Here is the solution from his solutions book: This is barely a proof. How can he just say let $f(c) = 0$? How do you prove that $f(c) =0$ and how do you prove that $f(d) ...
1
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0answers
66 views

Prove that if $T$ is one-to-one on $D$, then the set $T(D)$ is open

Let $f$ and $g$ have continuous first-order partial derivatives on an open set $ D\subseteq\mathbf{R}^2 $ and let $T :D \to \mathbf{R}^2 $ be defined by $ T(u,v)=(f(u,v),g(u,v)). $ ...
0
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1answer
23 views

Multiplication of two asymptotic expansions

I have two functions $g, f:(0,\infty)\rightarrow \mathbb{R}$ with asymptotic power series as follows: For all $N\in\mathbb{N}:$ $$f(t) \sim \sum\limits_{n=0}^{N} a_n t^n + O(t^{N+1}) \text{ }\text{ ...
0
votes
0answers
57 views

How to prove $f(c) = f(d) = 0$ [duplicate]

I ONLY NEED HELP WITH PROVING: $f(c) = f(d) = 0$ Robert Green's answer here : Is the first part of the answer, but I cannot problem that $f(c) = 0$ and $f(d) = 0$? How should I do this? Here was ...
1
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1answer
32 views

Constructing a Continuous Everywhere but Nowhere Differentiable Function

In Neal Carothers' Real Analysis he claims that $$f(x)=\sum_{k \mathop = 0}^\infty 2^{-k}g(2^{k}x)$$ is a continuous but non-differentiable function over the real line if $g(x)$ is the distance ...
1
vote
2answers
34 views

Differential calculus, curves and gradients $f'(-4) > f'(6)$ [closed]

Based on the graph below, is the statement $f'(-4) > f'(6)$ correct? And what are the approximate values?
2
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0answers
34 views

Positivity of the Fourier transform of a certain function

I am trying to show that the Fourier transform of $\cosh(x)^{-\nu}$ is positive for every $\nu>1$. I know that such a function has even Fourier transform and... that's about it. Could you suggets ...
0
votes
1answer
27 views

Metric triangle inequality $d_2(x,y):= \frac{d(x,y)}{d(x,y)+1}$

$(X,d)$ is a metric space. $x,y,z \in X$ Now I have to proof that $(X,d_2)$ is also a metric space. To show that $d_2(x,y)=0 \leftrightarrow x=y $ and $d_2(x,y) = d_2(y,x)$ are correct was quite ...
1
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2answers
39 views

A projection $P$ is orthogonal if and only if its spectral norm is 1

I have to show what the title says. A projection $P$ is orthogonal if and only if its spectral norm is $1$. I suppose I have to use the following identity: ...
0
votes
0answers
24 views

Property of $ L^p( [0,T] , X) $ with X Banach space

I'm starting to study the theory on linear parobolic equation using Evans's book "Partial differential equation". At pag 285, he speaks abaut $ L^p( [0,T] , X) $ with X Banach space. This type of ...
1
vote
1answer
47 views

Show that $\lim_{n \to +\infty} \frac{\sum^n x_i}{\sum^n y_i}=a.$ [duplicate]

Let $\{y_n\}$ a sequence so that $\sum y_i=+\infty$ and $$y_n>0, \forall n \in \mathbb{N}.$$ Show that if $$\lim_{n \to +\infty} \frac{x_n}{y_n}=a$$ then $$\lim_{n \to +\infty} ...
1
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2answers
36 views

Convergence and limits of recusive sequences.

I want to ask a question about recursive sequences. They have been pretty easy to handle for me, if you have one variable in it. To give you an example, if you have a sequence like: $x_o = 1, ...
2
votes
1answer
18 views

Difficult Limit involving digamma function

Evaluate: $$\lim_{z \to 0} \psi(-z)\cdot \bigg ( 1 - 2z(z+1) \bigg) - z\cdot\psi'(-z) $$ If we simply substitute in $0$ that gets us infinity, and problems. The answer is $-2 - \gamma$ How do we ...
0
votes
0answers
23 views

Proof that pull-back of closed set by continuous functional is weakly closed needs continuity?

Question: In Banach space $X$, if $\phi \in X^*$, then pullback of a closed set is weakly closed? I wrote the following proof: Let $X$ be Banach space. Let $I$ be a closed interval of ...
0
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1answer
44 views
0
votes
1answer
54 views

Norm of bounded real function

Let $X=[0,1]$ and $f$ be a continuous real function defined on $X$. The norm of $f$ is defined by $\Vert f\Vert$=sup$\vert f(x)\vert$ Pls. how do i show that the function $f$ is bounded and for $g$ ...
2
votes
1answer
38 views

Questions about sigma-algebra

I am learning measure theory this semester. The definition for sigma-algebra is "a collection of sets that is closed under complements and countable unions and intersections." I wonder what does it ...
4
votes
1answer
101 views

Discontinuous Differentiable and One to One

If the derivative of a function (from $\mathbb{R} \rightarrow \mathbb{R}$) at a point $x_0$ is discontinuous, does that imply that the function is not one to one or injective in a neighborhood of ...
1
vote
0answers
10 views

What are Carnot groups?

I'm trying to learn the Pansu differentiability theorem and I need to know what Carnot groups are. Can someone please explain what Carnot groups are? An introductory reference would be greatly ...
1
vote
0answers
34 views

Some equality from analysis book?

Can anyone prove the following claim ? Assume that $u_1,u_2:\mathbb{R}^n\to\mathbb{C}$ are Schwartz functions. Prove that for any $\xi\in\mathbb{R}^n$, $\xi\neq 0$ the following equality is ...
1
vote
0answers
17 views

Question on Uniform Continuity

Is it generally true that all uniformly continuous bijections $f: X \to Y$, where $X$ and $Y$ are metric spaces, have uniformly continuous inverses? If not, then what would be a counterexample, and is ...
3
votes
2answers
73 views

The value of $\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos^{2} nx dx.$

Using the fact $\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos nx dx=0$ ,find the value of $$\lim_{n\to \infty}\int_{-\infty}^{\infty}f(x)\cos^{2} nx dx.$$ I tried through integrating by parts , ...